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NAG Toolbox: nag_interp_1d_ratnl (e01ra)
Purpose
nag_interp_1d_ratnl (e01ra) produces, from a set of function values and corresponding abscissae, the coefficients of an interpolating rational function expressed in continued fraction form.
Syntax
Description
nag_interp_1d_ratnl (e01ra) produces the parameters of a rational function
$R\left(x\right)$ which assumes prescribed values
${f}_{i}$ at prescribed values
${x}_{i}$ of the independent variable
$x$, for
$\mathit{i}=1,2,\dots ,n$. More specifically,
nag_interp_1d_ratnl (e01ra) determines the parameters
${a}_{j}$, for
$\mathit{j}=1,2,\dots ,m$ and
${u}_{j}$, for
$\mathit{j}=1,2,\dots ,m1$, in the continued fraction
where
and
such that
$R\left({x}_{i}\right)={f}_{i}$, for
$\mathit{i}=1,2,\dots ,n$. The value of
$m$ in
(1) is determined by the function; normally
$m=n$. The values of
${u}_{j}$ form a reordered subset of the values of
${x}_{i}$ and their ordering is designed to ensure that a representation of the form
(1) is determined whenever one exists.
The subsequent evaluation of
(1) for given values of
$x$ can be carried out using
nag_interp_1d_ratnl_eval (e01rb).
The computational method employed in
nag_interp_1d_ratnl (e01ra) is the modification of the Thacher–Tukey algorithm described in
Graves–Morris and Hopkins (1981).
References
Graves–Morris P R and Hopkins T R (1981) Reliable rational interpolation Numer. Math. 36 111–128
Parameters
Compulsory Input Parameters
 1:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

${\mathbf{x}}\left(\mathit{i}\right)$ must be set to the value of the $\mathit{i}$th data abscissa, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Constraint:
the ${\mathbf{x}}\left(i\right)$ must be distinct.
 2:
$\mathrm{f}\left({\mathbf{n}}\right)$ – double array

${\mathbf{f}}\left(\mathit{i}\right)$ must be set to the value of the data ordinate, ${f}_{\mathit{i}}$, corresponding to ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Optional Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

Default:
the dimension of the arrays
x,
f. (An error is raised if these dimensions are not equal.)
$n$, the number of data points.
Constraint:
${\mathbf{n}}>0$.
Output Parameters
 1:
$\mathrm{m}$ – int64int32nag_int scalar

$m$, the number of terms in the continued fraction representation of $R\left(x\right)$.
 2:
$\mathrm{a}\left({\mathbf{n}}\right)$ – double array

${\mathbf{a}}\left(\mathit{j}\right)$ contains the value of the parameter
${a}_{\mathit{j}}$ in
$R\left(x\right)$, for
$\mathit{j}=1,2,\dots ,m$. The remaining elements of
a, if any, are set to zero.
 3:
$\mathrm{u}\left({\mathbf{n}}\right)$ – double array

${\mathbf{u}}\left(\mathit{j}\right)$ contains the value of the parameter
${u}_{\mathit{j}}$ in
$R\left(x\right)$, for
$\mathit{j}=1,2,\dots ,m1$. The
${u}_{j}$ are a permuted subset of the elements of
x. The remaining
$nm+1$ locations contain a permutation of the remaining
${x}_{i}$, which can be ignored.
 4:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{n}}\le 0$. 
 ${\mathbf{ifail}}=2$

At least one pair of the values ${\mathbf{x}}\left(i\right)$ are equal (or so nearly so that a subsequent division will inevitably cause overflow).
 ${\mathbf{ifail}}=3$

A continued fraction of the required form does not exist.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
Accuracy
Usually, it is not the accuracy of the coefficients produced by this function which is of prime interest, but rather the accuracy of the value of
$R\left(x\right)$ that is produced by the associated function
nag_interp_1d_ratnl_eval (e01rb) when subsequently it evaluates the continued fraction
(1) for a given value of
$x$. This final accuracy will depend mainly on the nature of the interpolation being performed. If interpolation of a ‘wellbehaved smooth’ function is attempted (and provided the data adequately represents the function), high accuracy will normally ensue, but, if the function is not so ‘smooth’ or extrapolation is being attempted, high accuracy is much less likely. Indeed, in extreme cases, results can be highly inaccurate.
There is no builtin test of accuracy but several courses are open to you to prevent the production or the acceptance of inaccurate results.
1. 
If the origin of a variable is well outside the range of its data values, the origin should be shifted to correct this; and, if the new data values are still excessively large or small, scaling to make the largest value of the order of unity is recommended. Thus, normalization to the range $1.0$ to $+1.0$ is ideal. This applies particularly to the independent variable; for the dependent variable, the removal of leading figures which are common to all the data values will usually suffice. 
2. 
To check the effect of rounding errors engendered in the functions themselves, nag_interp_1d_ratnl (e01ra) should be reentered with ${x}_{1}$ interchanged with ${x}_{i}$ and ${f}_{1}$ with ${f}_{i}$, $\left(i\ne 1\right)$. This will produce a completely different vector $a$ and a reordered vector $u$, but any change in the value of $R\left(x\right)$ subsequently produced by nag_interp_1d_ratnl_eval (e01rb) will be due solely to rounding error. 
3. 
Even if the data consist of calculated values of a formal mathematical function, it is only in exceptional circumstances that bounds for the interpolation error (the difference between the true value of the function underlying the data and the value which would be produced by the two functions if exact arithmetic were used) can be derived that are sufficiently precise to be of practical use. Consequently, you are recommended to rely on comparison checks: if extra data points are available, the calculation may be repeated with one or more data pairs added or exchanged, or alternatively, one of the original data pairs may be omitted. If the algorithms are being used for extrapolation, the calculations should be performed repeatedly with the $2,3,\dots \text{}$ nearest points until, hopefully, successive values of $R\left(x\right)$ for the given $x$ agree to the required accuracy. 
Further Comments
The time taken by nag_interp_1d_ratnl (e01ra) is approximately proportional to ${n}^{2}$.
The continued fraction
(1) when expanded produces a rational function in
$x$, the degree of whose numerator is either equal to or exceeds by unity that of the denominator. Only if this rather special form of interpolatory rational function is needed explicitly, would this function be used without subsequent entry (or entries) to
nag_interp_1d_ratnl_eval (e01rb).
Example
This example reads in the abscissae and ordinates of $5$ data points and prints the arguments ${a}_{j}$ and ${u}_{j}$ of a rational function which interpolates them.
Open in the MATLAB editor:
e01ra_example
function e01ra_example
fprintf('e01ra example results\n\n');
x = [0:4];
f = [4 2 4 7 10.4];
[m, a, u, ifail] = e01ra( ...
x, f);
disp('Reorder subset, u:');
disp(u(1:m1)');
fprintf('\n');
disp('Thiele coefficients, a:');
disp(a(1:m)');
e01ra example results
Reorder subset, u:
0 3 1
Thiele coefficients, a:
4.0000 1.0000 0.7500 1.0000
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